To start with, a broker is an intermediary between you and the liquid forex (foreign exchange) market. They allow you to make trades which can be thought of as bets on which way the market can go.
In a perfect world (or if you have a huge bank account), you would just place your order with the bank and pay a small fee. However, in the retail forex market with accounts under $50,000 you have 2 choices: "bucket shops" (aka: dealing desk) and non-dealing desk.
How these 2 classifications are generally explained is that with dealing desk brokers, your orders are routed through a desk and with with non-dealing desk brokers they go straight through to the "interbank liquidity market".
However, in the real world there is no time for the order to go through the "desk" as the prices are changing almost every second. In actuality how it works is that the "bucket shop" brokers use the fact that prices are changing quickly to make money from you. The reason they do this is because of the fact that over 90% of traders lose money. It is not in their best interests to pass your order through the bank if the odds are that you will lose so they take the other end of your trade and make money from your losses.
This is why scalping (very fast trades) is extremely difficult with a "bucket shop" broker. Let's say that you are getting in at a great time and you click buy, usually they will delay the price by a few seconds so that you lose the advantage that you would have had if you got the price you wanted. These brokers are also very difficult to trade using automated trading systems, as when the expert advisor tries to enter or exit your trade the price can be rejected.
Non-dealing desk brokers sound great in comparison. They say that there is "straight-through processing" so that the price is not interfered with. For this type of broker, usually you pay a commission each time you trade (plus a spread) which adds up substantially if you make a lot of trades each month. Also, in most cases the spreads are not fixed and they often change every second. The other factor is that they do not guarantee the price you click is the price you will get. Many times these brokers will advertise very low spreads and then you will see they are not that different from the bucket shop brokers after they widen unexpectantly and you get a pip or two of slippage.
Most of what I have explained will not affect you too much if you are trading on the higher time frames as a pip or two won't make a difference, but if you are scalping or intraday trading it will definitely affect your bottom line.
This being said, there are a handful (or less) of quality brokers, you just need to do your homework.